2023 KIAS HCMC
Algebraic Geometry Seminar

Previous Talks

November 29 (Wed) 2023, 10:30~12:00 - HCMC Seminar Room (수림문화재단 2F) 

Speaker: Junho Lee (University of Central Florida)
Title: A spin analog of GW/H correspondence

One of the beautiful results of the Gromov-Witten theory is the GW/H correspondence developed by Okounkov and Pandharipande, which relates GW invariants of curves and Hurwitz numbers. In this talk, I will discuss a conjectural formula for the spin analog of GW/H correspondence that relates GW invariants of Kähler surfaces and spin Hurwitz numbers – Giacchetto et al. recently proved the genus zero case.

November 13 (Mon) 2023, 11:00~12:00 - HCMC Seminar Room (수림문화재단 2F) 

Speaker: In-Kyun Kim (KIAS)
Title: Wall-crossing of K-moduli spaces of weighted projective spaces

In algebraic geometry, constructing moduli spaces that parametrize families of algebraic varieties with certain properties is an important problem. In the case of Fano varieties, the construction of moduli spaces is a challenging problem. K-stability provides a criterion for selecting nice representatives within the moduli space, which helps create more meaningful and well-behaved moduli spaces. In this talk, we will study some properties of K-moduli spaces.

October 16 (Mon) 2023, 10:30~12:00 - HCMC Seminar Room (수림문화재단 2F) 

Speaker: Sungwoo Nam (POSTECH)
Title: Simple normal crossing surfaces and local GW theory

In this talk, I’ll explain a definition of local Gromov-Witten (GW) invariant of some simple normal crossing surfaces, namely shrinkable surfaces. This will be a generalization of the local theory of smooth del Pezzo surfaces. I'll begin by studying the embedding problem for shrinkable surfaces and then explain how the local theory computes the contribution to global invariants. This is based on physics of M-theory on a Calabi-Yau 3-fold containing a shrinkable surface and also on the conjectural connection between shrinkable surfaces and 3-fold canonical singularities. If time permits, I'll talk about open questions related to more exotic surfaces, which are expected to have BPS invariants in physics.

September 18 (Mon) 2023, 16:00~17:00 - KIAS 1423

Speaker: Sukmoon Huh (Sungkyunkwan University)
Title: Generalizing the logarithmic sheaves

Roughly speaking, the logarithmic sheaf is the sheaf of differential one-forms with logarithmic poles along a divisor on a smooth projective variety. The sheaf was used extensively for the study of Hodge theory of quasi-projective manifolds, by Deligne-Saito. On the other hand, there have been investigations on this sheaf in a view point of moduli problem, such as when can it be free or when can it be Torelli. Quite recently, there are several attempts to generalize the sheaf in two different ways: one is to replace the divisor by an arbitrary subvariety, and the other is to consider its subsheaf to contain the information about the singularity of the divisor. In this talk, we sketch our contribution to the second direction and suggest some of its applications. This is a joint work with S. Marchesi, J. Pons-Llopis and J. Vallès. 

August 29 (Tue) 2023, 16:00~17:00 - KIAS 1423

Speaker: George Harry Hitching (Oslo Metropolitan University)

Title: Secant loci on Quot schemes of torsion quotients of vector bundles over a curve

Given a curve C and a linear system ℓ on C, the secant locus V^{e-f}_e(ℓ) parametrises effective divisors of degree e which impose at most e-f conditions on ℓ. For E a vector bundle of rank r over C and V a subspace of H^0(C, E*), we define a determinantal subscheme Q^{e-f}_e(V) of Quot^e(E*) which generalises V^{e-f}_e(ℓ). We describe the Zariski tangent spaces of Q^{e-f}_e(V), and show that smoothness in general is not controlled by injectivity of a Petri map. We generalise the Abel-Jacobi map and the notion of linear series to the context of Quot schemes. We then set f=1 and show that for sufficiently general E and V, the locus Q^{e-1}_e(V) is either empty or of the expected dimension. When Q^{e-1}_e(V) has and attains expected dimension zero, we indicate how formulas of Oprea-Pandharipande and Stark can be used to enumerate Q^{e-1}_e(V).

Talk 1: August 21 (Mon) 2023, 16:00~17:00 - HCMC Seminar Room (수림문화재단 2F)
Talk 2: August 23 (Wed) 2023, 11:00~12:00 - HCMC Seminar Room (수림문화재단 2F)

Speaker: Wei-Ping Li (Hong Kong University of Science and Technology)

Title: Gromov-Witten invariants of Calabi-Yau quintics

In the two talks, I will review the known results of Gromov-Witten invariants of Calabi-Yau quintics. I will also discuss about several key conjectures on higher genus Gromov-Witten invariants of Calabi-Yau quintics such as Bershadsky-Cecotti-Ooguri-Vafa’s conjecture and Yamaquchi-Yau finite generation ring. Then I will present a method developed by us, called mixed-spin-P-fields, which was used to solve the conjectures mentioned above. It is a joint work with Chang, Guo, J. Li, and Liu.

July 20 (Thu) 2023, 16:00~17:00 - KIAS 1423

Speaker: Insong Choe (Konkuk University)

Title: Simplicity of tangent bundles on the moduli of symplectic and orthogonal bundles

In this talk, we show the simplicity of the tangent bundles on the moduli of symplectic and orthogonal bundles over a curve. To prove this, we use the symplectic and orthogonal Hecke curves which are known to have minimal degree among the rational curves passing through a general point of the moduli space. The key idea is to check the non-degeneracy of the associated VMRT. This is based on a joint work with Jaehyun Hong and George H. Hitching.

June 19 (Mon) 2023, 11:00~12:00 - KIAS 1424

Speaker: Myeongjae Lee (Stony Brook University)

Title: Connected components of strata of residueless meromorphic differentials

Strata of differentials are interesting objected studied by various fields such as dynamics, topology and algebraic geometry. Generalized strata of meromorphic differentials are subsets of the usual strata of differentials, where certain sets of residues summing up to zero. They appear naturally in the boundary of the multi-scale compactification of the usual strata.  We classify connected components of the strata of residueless meromorphic differentials, which are the strata with maximal possible number of conditions imposed on the residues of the poles.

June 15 (Thu) 2023, 16:00~17:00 - KIAS 1424

Speaker: Yen-An Chen (National Center for Theoretical Sciences)

Title: MMP for toric foliations

In recent years, significant progress has been made in the field of birational geometry for foliations. Notably, the Minimal Model Program (MMP) has been shown to work for foliations on threefolds. In this talk, I will demonstrate that the MMP is applicable to toric foliations as well. Specifically, I will discuss how non-dicritical and F-dlt singularities are preserved under the MMP, respectively. This is a joint work by Chih-Wei Chang. 

June 14 (Wed) 2023, 16:00~17:30 - KIAS 1424

Speaker: In-Kyun Kim (Yonsei University)

Title: K-stability of ample divisors of del Pezzo surfaces

In Kahler geometry, searching for canonical metrics on a given Kahler manifold is an important problem. Proving the existence of the twisted Kahler-Einstein metric on a compact Kahler manifold is challenging, but recent progress has provided strong tools to address this problem. In this talk, we study how to prove the existence of the twisted Kahler-Einstein metrics on the del Pezzo surface of degree 5.

June 7 (Wed) 2023, 16:00~17:00 - KIAS 1424

Speaker: Hsin-Ku Chen (KIAS)

Title: Minimal resolutions of threefolds

We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities.

May 25 (Thu) 2023, 16:00~17:00 - KIAS 8406

Speaker: Jaewoo Jung (IBS-CCG)

Title: Bounds on Regularity of Monomial Ideals through Graph Decompositions

The Castelnuovo-Mumford regularity of varieties (or their defining ideals) is an algebraic invariant that encodes the algebraic complexity of the ideal. The regularity of monomial ideals can be studied using homologies of their associated simplicial complexes. We present a decomposition of the complex that bounds the regularity of the corresponding ideals. By combining this result with outcomes from structural graph theory, we improve or reprove known bounds on the regularity of monomial ideals.

May 10 (Wed) 2023, 14:00~15:00 - KIAS 1424

Speaker: Yaoxiong Wen (KIAS)

Title: Mirror symmetries for parabolic Hitchin systems, from classical to global, II

In the second talk, I will focus on the moduli space of parabolic Higgs bundles of type B and C. With the mirror pair of parabolic structures (or nilpotent orbits), I will briefly explain how to prove SYZ mirror symmetry and topological mirror symmetry. The main ingredient here is the local parabolic Higgs bundles which serve as a bridge between classical mirror symmetry and global mirror symmetry. This talk is based on the in-progress joint work with W. He, X. Su, B. Wang, and X. Wen.

May 8 (Mon) 2023, 10:00~11:00 - KIAS 1424

Speaker: Yaoxiong Wen (KIAS)

Title: Mirror symmetries for parabolic Hitchin systems, from classical to global, I

In the first talk, I will briefly review the Hitchin system's history and its mirror symmetries. Then mention our motivation for the parabolic Hitchin system. I will explain how the parabolic structures connect to nilpotent orbits. In the rest of the talk, I will explain the mirror symmetry for nilpotent orbit closures, i.e., the classical mirror symmetry. This talk is mainly based on the joint work with B. Fu and Y. Ruan (arXiv:2207.10533).

March 16 (Thu) 2023, 16:00~17:30 - KIAS 1424

Speaker: Sang-Bum Yoo (Gongju National University of Education)

Title: Higgs pairs and the Hitchin map over an irreducible nodal curve of arithmetic genus one

Higgs bundles on an elliptic curve were intensively studied by E. Franco, O. Garcia-Prada and P. E. Newstead. In this talk, we introduce their description of the moduli space of Higgs bundles on an elliptic curve and all fibers of the Hitchin map defined on the moduli space. Next, we describe the moduli space of Higgs pairs over an irreducible nodal curve of arithmetic genus one and all fibers of the Hitchin map on this moduli space by using the moduli space of generalized parabolic Hitchin pairs that U. N. Bhosle constructed.

February 27 (Mon) 2023, 16:00~17:30 - KIAS 1424

Speaker: Yoon-Joo Kim (ERC HyperK team, Universität Bonn)

Title: Isotrivial fibrations of hyper-Kähler fourfolds

A compact hyper-Kähler (HK) manifold and its Lagrangian fibration are higher-dimensional generalizations of a K3 surface and its elliptic fibration. A Lagrangian fibration f : X -> B of a HK manifold is called isotrivial if its smooth fibers are all isomorphic to each other. This is the most special type of Lagrangian fibrations, generalizing isotrivial elliptic fibrations of K3 surfaces. In this talk, I will propose a classification scheme of HK manifolds admitting isotrivial Lagrangian fibrations. The method in particular recovers the two well-known constructions of HK manifolds from scratch, the Hilbert schemes of n points on K3 surfaces and generalized Kummer varieties. I will then specialize the discussion to HK fourfolds. This is joint work with Radu Laza and Olivier Martin.

February 7 (Tue) 2023, 16:00~17:30 - KIAS 1424

Speaker: Kiryong Chung (Kyungpook National University)

Title: Rational curves in a quadric threefold via a SL(2,ℂ)-representation

In this talk, we regard the smooth quadric threefold Q3 as Lagrangian Grassmannian and search for fixed rational curves of low degree in Q3 with respect to a torus action, which is the maximal subgroup of the special linear group SL(2,ℂ). Most of them are confirmations of very well-known facts. If the degree of a rational curve is 3, it is confirmed using the Lagrangian's geometric properties that the moduli space of twisted cubic curves in Q3 has a specific projective bundle structure. If time is allowed, I will propose torus localizations of rational quartic curves space in other prime Fano threefolds. This is joint work with Sukmoon Huh and Sang-Bum Yoo.

Past Algebraic Geometry Seminars ...... Autumn 2016

Organizers
Hyunsuk Moon, Junho Choe, Hyeonjun Park, Jeong-Seop Kim
supported by June E Huh Center for Mathematical Challenges at Korea Institute for Advanced Study